Computational Geomechanics. Manuel Pastor

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is the velocity of the π phase relative to the solid.

      It is assumed that R^π is invertible, its inverse being and k^π is defined by the following relation:

(2.73)

where μ_π is the dynamic viscosity with dimensions [mass] [length]⁻¹ [time]⁻¹, K the intrinsic permeability [length]² and T the temperature above some datum. The permeability of (2.73) is the dynamic permeability [length]³ [mass]⁻¹ [time]; to obtain the soil mechanics permeability k′ [length]/[time], Equation (2.73) has to be multiplied by the specific weight of water γ_w of dimensions [mass] [length]⁻² [time]⁻². In the case of more than one fluid flowing, the intrinsic permeability is modified as

(2.74)

      where K^rπ is the relative permeability, a function of the degree of saturation. For the water density, the following holds:

(2.75)

      where K_w is the bulk modulus of water.

      2.5.5 General Field Equations

      The macroscopic balance laws are now transformed and the constitutive equations introduced, to obtain the general field equations.

The linear momentum balance equation for the fluid phases is obtained first. In Equation (2.65), the fluid acceleration is expressed, taking into account Equation (2.59), and introducing the relative fluid acceleration a^πs. Further, Equations (2.67) and (2.68) are introduced. The terms dependent on the gradient of the fluid velocity and the effects of phase change are neglected and a vector identity for the divergence of the stress tensor is used. Finally, Equations (2.73) and (2.74) are included, yielding

(2.76)

      The linear momentum balance equation for the solid phase is obtained in a similar way, taking into account Equations (2.68) instead of (2.67).

By summing this momentum balance equation with Equation (2.76) written for water and air and by taking into account the definition of total stress (2.69), assuming continuity of stress at the fluid‐solid interfaces and by introducing the averaged density of the multiphase medium

      (2.77)

      we obtain the linear momentum balance equation for the whole multiphase medium

(2.78)

      The mass balance equations are derived next.

The macroscopic mass balance equation for the solid phase (2.61), after differentiation and dividing by ρ^s is obtained as

(2.79)

This equation is used in the subsequent mass balance equations to eliminate the material time derivative of the porosity. For incompressible grains, as assumed here, . For compressible grains, see Equation (2.89) and related remarks.

The mass balance equation for water (2.63) is transformed as follows. First in Equation (2.75),

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